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Functional Differential Equations

Application of i-smooth calculus

  • Book
  • © 1999

Overview

Authors:
  1. A. V. Kim

    1. Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg, Russia

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Part of the book series: Mathematics and Its Applications (MAIA, volume 479)

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Table of contents (11 chapters)

  1. Front Matter

    Pages i-xv
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  2. i—Smooth Calculus

    1. Front Matter

      Pages 1-1
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    2. Structure of Functionals

      • A. V. Kim
      Pages 3-11

    3. Properties of Functionals. Invariant Derivative

      • A. V. Kim
      Pages 12-30

    4. Generalized Derivatives of Nonlinear Functionals

      • A. V. Kim
      Pages 31-38

  3. Functional Differential Equations

    1. Front Matter

      Pages 39-39
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    2. Functional Differential Equations

      • A. V. Kim
      Pages 41-59

    3. Neutral Functional Differential Equations

      • A. V. Kim
      Pages 60-67

  4. Direct Lyapunov Method for Systems with Delays

    1. Front Matter

      Pages 69-69
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    2. The Problem Statement

      • A. V. Kim
      Pages 71-73

    3. The Lyapunov Functional Method

      • A. V. Kim
      Pages 74-108

    4. The Lyapunov Function Method

      • A. V. Kim
      Pages 109-122

    5. Instability

      • A. V. Kim
      Pages 123-128

  5. Dynamical Programming Method for Systems with Delays

    1. Front Matter

      Pages 129-129
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    2. Systems with State Delays

      • A. V. Kim
      Pages 131-145

    3. Systems with Control Delays

      • A. V. Kim
      Pages 146-156

  6. Back Matter

    Pages 157-168
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Keywords

About this book

Beginning with the works of N.N.Krasovskii [81, 82, 83], which clari­ fied the functional nature of systems with delays, the functional approach provides a foundation for a complete theory of differential equations with delays. Based on the functional approach, different aspects of time-delay system theory have been developed with almost the same completeness as the corresponding field of ODE (ordinary differential equations) the­ ory. The term functional differential equations (FDE) is used as a syn­ onym for systems with delays 1. The systematic presentation of these re­ sults and further references can be found in a number of excellent books [2, 15, 22, 32, 34, 38, 41, 45, 50, 52, 77, 78, 81, 93, 102, 128]. In this monograph we present basic facts of i-smooth calculus ~ a new differential calculus of nonlinear functionals, based on the notion of the invariant derivative, and some of its applications to the qualitative theory of functional differential equations. Utilization of the new calculus is the main distinction of this book from other books devoted to FDE theory. Two other distinguishing features of the volume are the following: - the central concept that we use is the separation of finite dimensional and infinite dimensional components in the structures of FDE and functionals; - we use the conditional representation of functional differential equa­ tions, which is convenient for application of methods and constructions of i~smooth calculus to FDE theory.

Reviews

`The book is warmly recommended for students and researchers in mathematics, applied mathematicians, and engineers whose work involves ordinary differential equations, functional analysis, partial equations, optimal control and mathematics system theory.'
Acta Scientiarum Mathematicarum

Authors and Affiliations

  • Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg, Russia

    A. V. Kim

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